Evolution of zero-determinant strategy in iterated snowdrift game

Abstract

Zero-determinant strategy can set unilaterally or enforce a linear relationship on opponent's income, thereby achieving the purpose of blackmailing the opponent. So one can extort an unfair share from the opponent. Researchers often pay attention to the steady state and use the scores of the steady state in previous work. However, if the player changes his strategy frequently in daily game, the steady state cannot attain easily. It is necessary to attain the transient income if there is a difference in income between the previous state and the steady state. In addition, what will happen if evolutionary player encounters an extortioner? The evolutionary results cannot be proven, just using the simulations in previous work. Firstly, for the iterated game between extortioner and cooperator, we introduce the transient distribution, the transient income, and the arrival time to steady state by using the Markov chain theory. The results show that the extortioner's payoff in the previous state is higher than in the steady state when the extortion factor is small, and the results go into reverse when the extortion factor is large. Furthermore, the larger the extortion factor, the harder the cooperation will be. And the small extortion factor conduces to approaching the steady state earlier. The results provide a method to calculate the dynamic incomes of both sides and give us a time scale of reaching the steady state. Secondly, for the iterated game between extortioner and evolutionary player, we prove that the evolutionary player must evolve into a full cooperation strategy if he and his opponent are both defectors in the initial round. Then, supposing that the evolutionary speed is proportional to the gradient of his payoff, we simulate the evolutionary paths. It can be found that the evolutionary speeds are greatly different in four initial states. In particular, the evolutionary player changes his strategy into cooperation rapidly if he defects in the initial round. He also gradually evolves into a cooperator if he cooperates in the initial round. That is to say, the evolutionary process relates to his initial behavior, but the result is irrelevant to his behavior. It can be concluded that the zero-determinant strategy acts as a catalyst in promoting cooperation. Finally, we prove that the set of zero-determinant strategy and fully cooperation is not a Nash equilibrium.